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A259209
E.g.f.: Sum_{n>=1} x^(n^2) * exp(x^n) / n!.
5
1, 2, 3, 16, 5, 366, 7, 10088, 60489, 302410, 11, 89812812, 13, 363242894, 108972864015, 886312627216, 17, 178478870169618, 19, 101401086923136020, 354798209525760021, 1548722343168022, 23, 13787827750211997081624, 129260083694424883200025, 5051650697533440026
OFFSET
1,2
FORMULA
E.g.f.: -exp(1) + Sum_{n>=0} (1 + x^n)^n / n!.
a(n) = Sum_{d|n} binomial(d, n/d) * n!/d! for n>=1.
EXAMPLE
E.g.f.: A(x) = x + 2*x^2/2! + 3*x^3/3! + 16*x^4/4! + 5*x^5/5! + 366*x^6/6! +...
where
A(x) = x*exp(x) + x^4*exp(x^2)/2! + x^9*exp(x^3)/3! + x^16*exp(x^4)/4! + x^25*exp(x^5)/5! + x^36*exp(x^6)/6! +...
also
A(x) = -exp(1) + 1 + (1+x) + (1+x^2)^2/2! + (1+x^3)^3/3! + (1+x^4)^4/4! + (1+x^5)^5/5! + (1+x^6)^6/6! +...
Particular values.
A(1) = exp(2) - exp(1).
A(-1) = cosh(2) - exp(1).
A(1/2) = 0.8648559700938957468696599588156983897723576531...
A(1/2) = exp(1/2)/2 + exp(1/2^2)/(2!*2^4) + exp(1/2^3)/(3!*2^9) + exp(1/2^4)/(4!*2^16) + exp(1/2^5)/(5!*2^25) +...
A(1/2) = -exp(1) + 1 + (1+1/2) + (1+1/2^2)^2/2! + (1+1/2^3)^3/3! + (1+1/2^4)^4/4! + (1+1/2^5)^5/5! + (1+1/2^6)^6/6! +...
MATHEMATICA
Table[Sum[Binomial[d, n/d]*n!/d!, {d, Divisors[n]}], {n, 1, 30}] (* Vaclav Kotesovec, Oct 20 2020 *)
PROG
(PARI) {a(n) = my(A=1); A = sum(m=1, n, x^(m^2) * exp(x^m +x*O(x^n)) / m!); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); A = -exp(1) + sum(m=0, n, (1 + x^m +x*O(x^n))^m/m!); if(n==0, 0, n!*polcoeff(A, n))}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = if(n<1, 0, sumdiv(n, d, binomial(d, n/d) * n!/d! ) )}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 21 2015
STATUS
approved